Cycles does not show textures
On this vase you have three materials in the material slots tab. As you can see in your screenshot, you are editing the second one at the moment. Trouble is, that material (I know that from the last file you provided in another question Half of my mesh/texture is blurred) is not assigned to any face of your model, that's why you do not see the texture.

Either select all faces in Edit Mode and click the assign button in the material tab. This will assign the currently active material to the mesh faces. Or alternatively, delete the other materials (if you don't plan to use them). Then make sure the image texture node is selected in the Node Editor, and the 3D View is set to textured. That should do.